Review, due April 14, 2014

Which topics and theorems do you think are the most important out of those we have studied?

      I think that the Schroder-Bernstein Theorem.  The idea of implications and biconditionals are probably most important.  I like the fundamental theorem of arithmetic and the division algorithm.  I could see proofs by induction being very useful.

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

      I think just getting the calculus proofs down would be useful.  Anything will be helpful there.
What have you learned in this course? How might these things be useful to you in the future?

      I realize to a fuller extent, the importance of being specific and precise in language, particularly in math.  If we don't use precise language, we are prone to mistakes and problems, even if we don't foresee them.  I will be more precise in my language from now on.

12.5, due by April 10, 2014

      Continuity all seems to make sense.  It seems that the only addition here for proofs is to show that the limit exists and that the limit is equal to the value of the function at that point.  I think this section should be most straight forward so long as I can keep delta-epsilon proofs straight.
      This is a useful topic though because it connects limits with the actual function to see it the limit that we predict is included where we think it should be.  I wonder though how we would work off of limits approaching from one side or from another.

12.4, due by April 9, 2014

      Most of the proofs there seemed quite difficult to follow.  I think I need to go back to section 12.3 again to see if I really understand the proofs there.  They still feel a little complicated and I can't seem to keep my variables straight.  I'm sure it will work out after I've done the homework problems for section 12.3.
      This section makes proving the limits of many things so much easier.  I especially liked the proofs at the end of the section which proved generalized limits of polynomials by induction.  It just makes things so much easier.

12.3, due April 7, 2014

      I think delta-epsilon proofs make sense, they just take a little more getting used to with keeping track of the numbers.  As always, I will have to do plenty of practice problems.
      Back in calculus they felt backward for the reason that you can only do them if you have a sense of what the limit is and there never seemed much reason to prove it otherwise.  Though now it makes a lot more sense why we would want to approach limits from a delta-epsilon perspective.

12.2, due by April 4

      The most difficult part will probably be making a good guess for what the pattern is for the formula of the sum.  I will live in hopes that I will not be assigned problems that are too hard.  Otherwise it should work out.
      I enjoy the proofs about series.  I like the idea of making a lemma so that it just becomes a problem of finding the limit.  I find it impressive how there even exist methods to take series and convert them to a formula that can be solved so much easier to deal with.

12.1, due by April 2, 2014

      The most difficult part was keeping straight N, L, and epsilon and what the significance of each one.  When I walk myself through the proofs and keep all of those variables straight, the proof is apparent so I just need practice.
      I like the proofs in calculus.  It is interesting to see that the tools we have to work with can be applied to calculus very easily when given clear definitions.  I like the idea that these proofs can be done in terms of integers and such.

Review, due March 31

• Which topics and theorems do you think are the most important out of those we have studied?
• The Schroder-Bernstein Theorem
• The Fundamental Theorem of Arithmetic
• gcd
• The Division Algorithm
• Cardinality of denumerable and nondenumberable sets
• What kinds of questions do you expect to see on the exam?
• I expect to see a lot of definitions, and proofs
• Specific proofs I expect to see are ones using the Schroder-Bernstein theorem, which should be pretty straight forward.  I hope not to see one where I need create a bijective function that from a subset of the real numbers to the real numbers but I will be prepared.  I expect to see some questions on relative primes and the division algorithm.  It should be a fun exam.
• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.
• I need to understand problems that involve uncountable sets and using the decimal expansion to prove things.  I think 10.36 would help me understand things better.